Real exponential functions/Growing behavior/Remark

For a real exponential function

${}y(x)=a^{x}\,,$

the relation

${}y'={\left(\ln a\right)}y\,$

holds, due to fact. Hence, there is a proportional relationship between the function ${}y$ and its derivative ${}y'$, and ${}\ln a$ is the factor. This is still true if ${}a^{x}$ is multiplied with a constant. If we consider ${}y$ as a function depending on time ${}x$, then ${}y'(x)$ describes the growing behavior at that point of time. The equation ${}y'={\left(\ln a\right)}y$ means that the instantaneous growing rate is always proportional with the magnitude of the function. Such an increasing behavior (or decreasing behavior, if ${}a<1$) occurs in nature for a population, if there is no competition for resources, and if the dying rate is neglectable (the number of mice is then proportional with the number of mice born). A condition of the form

${}y'=by\,$

is an example of a differential equation. This is an equation for a function, which expresses a condition for the derivative. A solution for such a differential equation is a differentiable function which fulfills the condition on its derivative. The differential equation just mentioned are fulfilled by the functions

${}y(x)=ce^{bx}\,.$